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Cited by 646Thermal properties of materialsThermography and Photoacoustic TechniquesThermoelastic and Magnetoelastic Phenomena

Broadband phonon mean free path contributions to thermal conductivity measured using frequency domain thermoreflectance

Open access
Regner, Keith T., Sellan, Daniel P., Su, Zonghui, Amon, Cristina H., McGaughey, Alan J.H., Malen, Jonathan A.·2013·Nature Communications
DOI 10.1038/ncomms2630
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Abstract

Non-metallic crystalline materials conduct heat by the transport of quantized atomic lattice vibrations called phonons. Thermal conductivity depends on how far phonons travel between scattering events-their mean free paths. Due to the breadth of the phonon mean free path spectrum, nanostructuring materials can reduce thermal conductivity from bulk by scattering long mean free path phonons, whereas short mean free path phonons are unaffected. Here we use a breakdown in diffusive phonon transport generated by high-frequency surface temperature modulation to identify the mean free path-dependent contributions of phonons to thermal conductivity in crystalline and amorphous silicon. Our measurements probe a broad range of mean free paths in crystalline silicon spanning 0.3-8.0 μm at a temperature of 311 K and show that 40±5% of its thermal conductivity comes from phonons with mean free path >1 μm. In a 500 nm thick amorphous silicon film, despite atomic disorder, we identify propagating phonon-like modes that contribute >35±7% to thermal conductivity at a temperature of 306 K.

AI snapshot

Written from the paper’s full text.

TL;DR

Broadband frequency domain thermoreflectance (BB-FDTR) measures phonon mean free path (MFP) contributions to thermal conductivity by varying the thermal penetration depth with heating frequency. In crystalline silicon at 311 K, phonons with MFPs longer than 1 μm contribute 40±5% of bulk thermal conductivity, and in a 500 nm amorphous silicon film, propagating phonon-like modes with MFPs >100 nm contribute 35±7%.

Key findings

  • Intrinsic c-Si probed over L_p from 0.3–8.0 mm (as printed) at 311 K: phonons with MFPs longer than 1 mm contribute 40±5% to bulk thermal conductivity; 95±6% of bulk conductivity is recovered at the lowest heating frequency (200 kHz, L_p ≈ 8 mm).
  • A constant-thermal-conductivity fit for intrinsic c-Si over the full frequency range gives 99±6 W m⁻¹ K⁻¹, below the bulk value of 143 W m⁻¹ K⁻¹; window-fitting reveals a frequency-dependent thermal conductivity.
  • SiO₂ shows frequency-independent thermal conductivity of 1.4±0.2 W m⁻¹ K⁻¹, consistent with energy-carrier MFPs shorter than 60 nm; Pt also shows no L_p dependence because electrons with MFPs ≈10 nm dominate and are strongly coupled to the lattice.
  • The 500 nm a-Si film shows 35±7% of its thermal conductivity from propagating phonon-like modes with MFPs longer than 100 nm; the spectrum is flat below 100 nm, transitions steeply from 100–500 nm, and asymptotes as L_p exceeds the 500 nm film thickness.

Methods

  • BB-FDTR uses a 488 nm CW pump modulated from 200 kHz to 200 MHz and a 532 nm CW probe; a heterodyne scheme with a second electro-optic modulator at f₂ shifts the signal to f₁−f₂ = 86 kHz to reject ambient and coherent noise. Phase-lag data are taken at 60 frequencies, and frequency-dependent fits use overlapping windows of 13 points.
  • Samples include intrinsic c-Si, doped c-Si (3×10¹⁹ cm⁻³ boron), thermally grown SiO₂, single-crystal Pt, and sputtered a-Si films (500 nm, 1 μm, 2 μm), all coated with Cr/Au transducers. Phase-lag data are fit to an analytical diffusive heat-conduction solution, with sample thermal conductivity and Au-Cr/sample interface conductance G as fitting parameters.
  • A lattice Boltzmann method solves the one-dimensional phonon Boltzmann transport equation under the relaxation-time approximation and grey approximation to model the diffusive-to-ballistic transition for an FDTR-like system.

Limitations

  • Stated assumptions: the transducer/sample interface thermal conductance and the volumetric heat capacity are both assumed to be modulation-frequency independent; the authors argue these are justified by fast electron relaxation in Au and the negligible heat-capacity contribution of long-MFP phonons.
  • Stated experimental limits: increasing frequency beyond 20 MHz causes signal-to-noise issues because the temperature response amplitude decreases while coherent and ambient noise increase, which the heterodyne approach is designed to mitigate.
  • Evident from the text: the MFP spectra rely on interpreting the effective diffusive-fit thermal conductivity as k_accum with L* = L_p; the BTE modeling uses a grey approximation with a single group velocity and MFP; and the a-Si spectrum is normalized using the maximum measured value rather than a bulk value.

Model-written summary. Check it against the paper before relying on it.

Integrity signals

Registry

Crossref · 10.1038/ncomms2630

Cites

48 of 48 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Claims the paper makes

5 of 30 cited sources checked · 4 supported · 0 flagged · accuracy not yet measured

  • Findingown data

    Our data compare favourably with the first principles predictions by Esfarjani and coworkers at all temperatures.

    “Although our data compare favourably with the first principles predictions by Esfarjani and coworkers 1,2,41 at all temperatures”
    • Also stated in: § BTE solution methodology. · plain
      “Comparison with predictions from Minnich et al. 1 and Esfarjani and Chen 2 are favourable at all temperatures.”
    Quote in the paperNot a substantial claim2 of 3 sources checked
    • cites [1] Minnich, A. J. et al. Thermal conductivity spectroscopy technique to measure phonon mean free paths. Phys. Rev. Lett. 107, 095901 (2011). source
    • cites [2] Esfarjani, K. & Chen, G. . Heat transport in silicon from first-principles calculations. Phys. Rev. B 84, 085204 (2011). source
    • cites [41] Esfarjani, K. & Stokes, H. T. . Method to extract anharmonic force constants from first principles calculations. Phys. Rev. B 77, 144112 (2008). source
  • Findingown data

    In intrinsic c-Si, 95 ± 6% of the bulk thermal conductivity is obtained at the lowest heating frequency of 200 kHz, corresponding to a thermal penetration depth of approximately 8 μm.

    “We also note that 95 ± 6% of the bulk thermal conductivity is obtained at the lowest heating frequency (200 kHz, L p E8 mm).”
    Quote in the paperSubstantial claim
  • Findingown data

    Window-fitting the phase data for intrinsic c-Si yields a frequency-dependent thermal conductivity.

    “As shown in Fig. 1b, a frequencydependent thermal conductivity is obtained.”
    Quote in the paperSubstantial claim
  • Findingown data

    Window-fitting the phase data for SiO2 yields a frequency-independent thermal conductivity.

    “Window-fitting the phase data for SiO 2 , however, yields a frequency-independent thermal conductivity.”
    Quote in the paperNot a substantial claim
  • Findingown data

    In SiO2, thermal penetration depths from 60 to 900 nm yield a constant value of thermal conductivity.

    “In SiO 2 , L p,i from 60-900 nm yield a constant value of thermal conductivity”
    Quote in the paperSubstantial claim
  • Classified as probably not a substantial claim. 9
  • May combine two claims. 13
  • The quote only partly states this claim. 2

  • Classified as probably not a substantial claim. May combine two claims.

    Premisecitation

    The long mean free path phonons probed by BB-FDTR contribute substantially to thermal conductivity, but not to heat capacity (less than 1%).

    “Regarding (ii), the long MFP phonons probed by BB-FDTR contribute substantially to thermal conductivity, but not to heat capacity (o1%) 2 .”
    Quote in the paperSubstantial claim1 of 1 sources checked
    • cites [2] Esfarjani, K. & Chen, G. . Heat transport in silicon from first-principles calculations. Phys. Rev. B 84, 085204 (2011). source

      Section E, “Thermal conductivity from lattice dynamics” (page not available), reports that phonons with mean free paths longer than 1 micron contribute almost half of silicon’s thermal conductivity. The cited paper does not establish that those phonons contribute less than 1% of heat capacity or identify them as phonons probed by BB-FDTR. (LLM, conf=0.86)

      Experimental — accuracy not yet measured

  • May combine two claims.

    Premisecitation

    This interpretation is the same as that of Koh and Cahill and consistent with Minnich et al. and Johnson et al., who instead used the laser spot diameter and transient grating period as the cutoff dimensions.

    “This interpretation is the same as that of Koh and Cahill 16 , and consistent with Minnich et al. 1 and Johnson et al. 19,20 who instead used the laser spot diameter and transient grating period as the cutoff dimensions.”
    Quote in the paperNot a substantial claim2 of 4 sources checked
    • cites [16] Koh, Y. K. & Cahill, D. G. . Frequency dependence of the thermal conductivity of semiconductor alloys. Phys. Rev. B 76, 075207 (2007). source
    • cites [1] Minnich, A. J. et al. Thermal conductivity spectroscopy technique to measure phonon mean free paths. Phys. Rev. Lett. 107, 095901 (2011). source
    • cites [19] Johnson, J. A. et al. Direct measurement of room-temperature nondiffusive thermal transport over micron distances in a silicon membrane. Phys. Rev. Lett. 110, 025901 (2013). source
    • cites [20] Johnson, J. A. et al. Experimental evidence of non-diffusive thermal transport in Si and GaAs. MRS Proc. 1347, doi:10.1557/opl.2011.1333 (2011). source

  • May combine two claims.

    Findingown data

    In intrinsic c-Si, fitting phase-lag data over the entire frequency range to a constant thermal conductivity yields 99 ± 6 W m−1 K−1, which underpredicts the bulk value of 143 W m−1 K−1 and has a large mean square error of 2.4.

    “In intrinsic c-Si, however, the constant thermal conductivity fit underpredicts the bulk value (143 W m À 1 K À 1 (ref.25)) as 99±6 W m À 1 K À 1 with a large MSE of 2.4.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [25] Inyushkin, A. V., Taldenkov, A. N., Gibin, A. M., Gusev, A. V. & Pohl, H. J. . On the isotope effect in thermal conductivity of silicon. Phys. Status Solidi C 1, 2995–2998 (2004). source

  • Probably not a substantial claim: may not pass S1 (asserted), S3 (load-bearing), S4 (based).

    Findingown data

    In intrinsic c-Si at 311 K, phonons with mean free paths longer than 1 μm contribute 40 ± 5% to the bulk thermal conductivity.

    “In intrinsic c-Si, we probe L p from 0.3-8.0mm and find that phonons with MFPs longer than 1 mm contribute 40±5% to the bulk thermal conductivity.”
    • Also stated in: § BTE solution methodology. · plain
      “At T ¼ 300 K, phonons with MFP41 mm contribute 40±5% to bulk thermal conductivity.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingown data

    For SiO2, fitting phase-lag versus modulation frequency data over the entire frequency range to a constant thermal conductivity yields 1.4 ± 0.2 W m−1 K−1, which agrees with bulk values and has a low mean square error of 0.04.

    “For SiO 2 , this fit yields a thermal conductivity of 1.4 ± 0.2 W m À 1 K À 1 , which agrees with bulk values 24 and has a low mean square error (MSE ¼ 0.04).”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [24] Yamane, T., Nagai, N., Katayama, S. & Todoki, M. . Measurement of thermal conductivity of silicon dioxide thin films using a 3ω method. J. Appl. Phys. 91, 9772–9776 (2002). source

  • The quote only partly states this claim.

    Findingown data

    In a 500 nm thick amorphous silicon film, 35 ± 7% of its thermal conductivity comes from propagating phonon-like modes with mean free paths longer than 100 nm.

    “The MFP spectrum of the 500 nm a-Si film shows that propagating phonons with MFPs 4100 nm contribute 35±7% to its thermal conductivity (for a-Si, we have used our maximum value for normalization).”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning

    The a-Si mean free path spectrum is flat below 100 nm as all propagating modes ballistically transmit through the domain, whereas the nonpropagating modes remain diffusive.

    “The spectrum is flat below 100 nm as all of the propagating modes ballistically transmit through the domain, whereas the nonpropagating modes remain diffusive.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning

    At high heating frequencies, the thermal penetration depth decreases below the mean free paths of some phonons, which travel ballistically through the thermally affected zone.

    “At high heating frequencies, the thermal penetration depth decreases below the MFPs of some phonons, which travel ballistically through the thermally affected zone (white arrows).”
    Quote in the paperSubstantial claim

  • The quote only partly states this claim.

    Findingreasoning

    At lower temperatures of 81 and 155 K, longer mean free path phonons contribute more significantly to the measured thermal conductivity.

    “At lower temperatures, longer MFP phonons contribute more significantly to the measured thermal conductivity.”
    Quote in the paperSubstantial claim

  • Classified as probably not a substantial claim.

    Premisecitation

    More recent simulations and experiments have also pointed to the existence of long mean free path phonon-like modes in a-Si.

    “More recent simulations and experiments have also pointed to the existence of long MFP phonon-like modes in a-Si 34,[36][37][38] .”
    Quote in the paperSubstantial claim0 of 3 sources checked
    • cites [34] Liu, X. et al. High thermal conductivity of a hydrogenated amorphous silicon film. Phys. Rev. Lett. 102, 035901 (2009). source
    • cites [36] He, Y., Donadio, D. & Galli, G. . Heat transport in amorphous silicon–interplay between morphology and disorder. Appl. Phys. Lett. 98, 144101 (2011). source
    • cites 37][38

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Study design and conductcitation

    The associated values of G are consistent with prior measurements of a Cr/Si interface.

    “The associated values of G, shown in the inset, are consistent with prior measurements of a Cr/Si interface 39 .”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [39] Stevens, R. J., Smith, A. N. & Norris, P. M. . Measurement of thermal boundary conductance of a series of metal-dielectric interfaces by the transient thermoreflectance technique. J. Heat Transfer 127, 315–322 (2005). source

  • May combine two claims.

    Premisecitation

    At temperatures of 81 and 155 K, the heat capacity is at 20 and 45% of saturation, respectively; high-frequency phonon modes are not populated, and phonon-phonon scattering is reduced relative to 311 K.

    “At temperatures of 81 and 155 K, the heat capacity is at 20 and 45% of saturation, respectively 40 .High-frequency phonon modes are not populated, and phonon-phonon scattering is reduced relative to T ¼ 311 K.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [40] Desai, P. D. . Thermodynamic properties of iron and silicon. J. Phys. Chem. Ref. Data 15, 967–983 (1986). source

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Study design and conductcitation

    Uncertainty displayed as shaded regions was calculated using the procedure in Malen et al. by considering uncertainty in the Au-Cr transducer thickness, the Au-Cr transducer thermal conductivity, G, spot radius, and the propagation of error in the measured phase.

    “Uncertainty, displayed as shaded regions in Figs 3 and 4, was calculated using the procedure in Malen et al. 42 by considering uncertainty in the Au-Cr transducer thickness (±5%), the Au-Cr transducer thermal conductivity (±5%), G (sample dependant), spot radius (3.4±0.1 mm) and the propagation of error in the measured phase (see Supplementary Note 2).”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [42] Malen, J. A. et al. Optical measurement of thermal conductivity using fiber aligned frequency domain thermoreflectance. J. Heat Transfer 133, 081601 (2011). source

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation

    Propagating phononlike modes in a-Si were first inferred by a disagreement between thermal conductivity measurements and modelling predictions that was resolved by the consideration of low-frequency Debye modes.

    “Propagating phononlike modes in a-Si were first inferred by a disagreement between thermal conductivity measurements and modelling predictions that was resolved by the consideration of low-frequency Debye modes 35 .”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [35] Cahill, D. G., Katiyar, M. & Abelson, J. R. . Thermal conductivity of a-Si:H thin films. Phys. Rev. B 50, 6077–6081 (1994). source

  • May combine two claims.

    Interpretationreasoning

    Similar features exist in thicker a-Si films, though the transition is delayed to larger Lp, indicating that k_accum is influenced by the a-Si film boundaries.

    “Similar features exist in thicker films, though the transition is delayed to larger L p , indicating that k accum is influenced by the a-Si film boundaries (see Supplementary Discussion).”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

    Interpretationreasoning

    BB-FDTR's continuous resolution of the mean free path spectrum indicates how the dissipative ability of c-Si will continuously decrease as transistor size decreases to uphold Moore's law.

    “BB-FDTR's continuous resolution of the MFP spectrum indicates how the dissipative ability of c-Si will continuously decrease as transistor size decreases to uphold Moore's law.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Interpretationnone given

    Our general experimental approach can be applied to measure the phonon mean free path spectra of other semiconducting and insulating solids.

    “Our general experimental approach can be applied to measure the phonon MFP spectra of other semiconducting and insulating solids.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Premisecitation

    Using atomistic calculations, He et al. define modes with mean free paths less than 10 nm as diffusons, which contribute approximately half to the bulk thermal conductivity.

    “Using atomistic calculations, He et al. 36 define modes with MFPs o10 nm as diffusons, which contribute approximately half to the bulk thermal conductivity.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [36] He, Y., Donadio, D. & Galli, G. . Heat transport in amorphous silicon–interplay between morphology and disorder. Appl. Phys. Lett. 98, 144101 (2011). source

  • May combine two claims.

    Study design and conductown data

    Low angle X-ray diffraction shows no crystalline peaks and confirms that the a-Si films are amorphous.

    “Low angle X-ray diffraction shows no crystalline peaks and confirms that the a-Si films are amorphous (see Supplementary Note S4).”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning

    At low heating frequencies, when the thermal penetration depth is greater than the phonon mean free paths, there is diffusive thermal transport, and a bulk value of thermal conductivity is measured.

    “At low heating frequencies, when the thermal penetration depth is greater than the phonon MFPs, there is diffusive thermal transport, and a bulk value of thermal conductivity is measured.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationown data

    BB-FDTR can aid in understanding thermal transport in amorphous solids by identifying propagating phonon modes and generating thermal conductivity accumulation functions.

    “Likewise, BB-FDTR can aid in the understanding of thermal transport in amorphous solids by identifying propagating phonon modes and generating thermal conductivity accumulation functions.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Premisecitation

    In platinum, electrons with mean free paths of about 10 nm are the dominant heat carriers, and strong electron-phonon coupling ensures that they are in thermal equilibrium with the lattice.

    “In Pt, electrons with MFPs B10 nm (ref.32) are the dominant heat carriers, and strong electron-phonon coupling ensures that they are in thermal equilibrium with the lattice 33 .”
    Quote in the paperSubstantial claim0 of 2 sources checked
    • cites [32] Fischer, G., Hoffmann, H. & Vancea, J. . Mean free path and density of conductance electrons in platinum determined by the size effect in extremely thin films. Phys. Rev. B 22, 6065–6073 (1980). source
    • cites [33] Wang, W. & Cahill, D. G. . Limits to thermal transport in nanoscale metal bilayers due to weak electron-phonon coupling in Au and Cu. Phys. Rev. Lett. 109, 175503 (2012). source
22 of 22

  • The authors state this as an assumption.

    The transducer/sample interface thermal conductance is modulation frequency independent.

    “Two assumptions are critical to the MFP spectra generated using this approach: (i) the transducer/sample interface thermal conductance is modulation frequency independent and (ii) the volumetric heat capacity is modulation frequency independent.”
    “In both cases, G is assumed to be frequency independent (see Methods).”

  • The authors state this as an assumption.

    The volumetric heat capacity is modulation frequency independent.

    “Two assumptions are critical to the MFP spectra generated using this approach: (i) the transducer/sample interface thermal conductance is modulation frequency independent and (ii) the volumetric heat capacity is modulation frequency independent.”

  • The authors state this as an assumption.

    All phonons have the same group velocity and mean free path, that is, the grey approximation.

    “All phonons are assumed to have the same group velocity and MFP (that is, the grey approximation).”
3 of 3

Claims other papers make about this one

Each is checked against this paper’s own text: does it say what the citing paper says it does?

  • A constant thermal conductivity over all heating frequencies under predicts thermal conductivity in single crystal Si and in the GaAs, GaN, AlN, and 4H-SiC crystals studied here.

    “On the other hand, a constant thermal conductivity over all heating frequencies was found to under predict thermal conductivity in single crystal Si1, and likewise in the GaAs, GaN, AlN, and 4H-SiC crystals studied here.”

    From Universal phonon mean free path spectra in crystalline semiconductors at high temperature (2013) · cites this paper as [1]

  • It is possible to experimentally measure the MFP dependent contributions of phonons to thermal conductivity (the phonon MFP spectrum).

    “Recent studies have shown that it is possible to experimentally measure the MFP dependent contributions of phonons to thermal conductivity–the phonon MFP spectrum12345.”

    From Universal phonon mean free path spectra in crystalline semiconductors at high temperature (2013) · cites this paper as [1]

  • Care should be taken when applying a diffusive thermal transport model to systems where a significant fraction of heat is carried by phonons with long mean-free paths (MFPs).

    “However, care should be taken when applying a diffusive thermal transport model to systems where a significant fraction of heat is carried by phonons with long mean-free paths (MFPs). 90-92, 112”

    From Tutorial: Time-domain thermoreflectance (TDTR) for thermal property characterization of bulk and thin film materials (2018) · cites this paper as [92]

  • BB-FDTR was introduced to study phonon mean free path spectra in semiconductors, similar to frequency-dependent TDTR based on the physical picture of non-diffusive phonon transport.

    “For example, it was introduced to study phonon MFP spectra in semiconductors, 92, 147, 148 similar to the frequency-dependent TDTR that is based on the physical picture of non-diffusive phonon transport, see more details in Section IV(D).”

    From Tutorial: Time-domain thermoreflectance (TDTR) for thermal property characterization of bulk and thin film materials (2018) · cites this paper as [92]

Cites

Flagged first, then by citations

Research works printed but not linked: 1

  1. Incropera, F. P., Dewitt, D. P., Bergman, T. L. & Lavine, A. S. . Fundamentals of Heat and Mass Transfer 6 edn John Wiley & Sons (2007).

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Related

Authors
Regner, K. T., Sellan, D. P., Su, Z., Amon, C. H., McGaughey, A. J. H., & Malen, J. A.
Affiliations
  • Carnegie Mellon University (US)
  • University of Toronto (CA)
Corresponding author
Malen, Jonathan A.
ORCID
Venue
Nature Communications
Publisher
Springer Science and Business Media LLC
Published
2013-03-27
Volume
4
Issue
1
Pages
1640-1640
ISSN
2041-1723 (ISSN-L)
DOI
10.1038/ncomms2630
OpenAlex
W2132162076
Semantic Scholar
20c913f5468edb5fc5cafcd323c3222e7669952a
Cited by
646
References
46
Citations by year
2026: 47 · 2025: 50 · 2024: 39 · 2023: 30 · 2022: 51 · 2021: 44 · 2020: 32 · 2019: 51 · 2018: 60 · 2017: 59 · 2016: 72 · 2015: 54 · 2014: 43 · 2013: 14
FWCI
27.07
Type
journal article
Language
en
Open access
Licence
Locations
  • Nature Communications (publishedVersion)
  • PubMed (publishedVersion)
  • RePEc: Research Papers in Economics (submittedVersion)
Funders
  • National Science Foundation (CAREER, 1149374)
  • Natural Sciences and Engineering Research Council of Canada
  • Air Force Office of Scientific Research (FA95501010098)
Topics
  • Physical Sciences › Materials Science › Materials Chemistry › Thermal properties of materials
  • Physical Sciences › Engineering › Mechanics of Materials › Thermography and Photoacoustic Techniques
  • Physical Sciences › Engineering › Mechanics of Materials › Thermoelastic and Magnetoelastic Phenomena
Keywords
phonon mean free path, thermal conductivity, frequency-domain thermoreflectance, phonon transport, crystalline silicon, amorphous silicon, nanostructuring, phonon scattering, lattice vibrations

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